Lens and optical system

ABSTRACT

A lens has a convex or concave shaped smooth non-spherical surface or non-circular curve which is formed of a medium indicating negative refraction.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is based upon and claims the benefit of priority from prior Japanese Patent Application No. 2005-123668, filed Apr. 21, 2005, the entire contents of which are incorporated herein by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a lens and an optical system including the lens, and an optical device including the lens.

2. Description of the Related Art

Conventionally, there have been well known: an optical elements that uses a light wave or electromagnetic wave; optical systems such as an image pickup optical system, an observation optical system, a projection optical system, and a signal processing system; and an optical device that uses these systems. These optical systems have a defect that image resolution is limited because of diffraction that occurs due to undulation property of a light wave or electromagnetic wave.

Therefore, as a technique for achieving image formation which exceeds the diffraction limit, use of a negative refractive index medium is disclosed in the following non-patent documents 2 and 5, etc.

FIG. 10 is a view for explaining an example of a technique for achieving such image formation. This figure shows image formation using a parallel flat plate 380 formed of a negative refractive index medium 301. In FIG. 10, t₀ denotes a distance between an object point and a left side face of the flat plate 380; t₀, denotes a distance between an image point and a right side face of the flat plate 380; “t” denotes a thickness of the flat plate 380; “i” denotes an incident angle; “r” denotes a refraction angle; and n_(s) denotes a refractive index of the negative refractive index medium 301 with respect to a vacuum.

A refractive index of the periphery of the flat plate 380 with respect to a vacuum is n₀, and n₀=1 is established in the case of a vacuum. FIG. 1 shows a case in which n₀=1 and n_(s)=−1 are established.

The arrow indicates an emitted light component from among the light beams emitted from an object. Because a refraction law is established according to non-patent document 2, the following formula is established: n ₀ sin i=n _(s) sin r  Formula 101

Assuming that n₀=1 and n_(s)=−1, the following formula is established: r=−i  Formula 102

Therefore, a light beam of the emitted light component is formed as an image point at a point at which t_(0′) satisfies the following formula: t ₀ +t _(0′) =t  Formula 103

On the other hand, an evanescent wave produced from an object point has intensity equal to that of the object point at a point at which t_(0′) satisfies Formula 103 as well. All the light beams emitted from the object point are collected at the image point, and thus, image formation exceeding the diffraction limit is achieved. This is referred to as complete image formation. It is known from non-patent document 2 listed below that, even if the periphery of the negative refractive index medium 301 is not a vacuum, complete image formation is achieved when Formula 103 and Formula 104 are satisfied: n _(s) =−n ₀  Formula 104

Non-patent document 1: Mechanism and application of optical system, 73-77, 166-170, Optronics Co., Ltd., 2003

Non-patent document 2: J. B. Pendry Phys. Rev. Lett., Vol. 85, 18 (2000) 3966-3969

Non-patent document 3: M. Notomi Phys. Rev. B. Vol. 62 (2000) 10696

Non-patent document 4: V. G. Veselago Sov. Phys. Usp. Vol. 10, 509-514 (1968)

Non-patent document 5: L. Liu and S. He Optics Express Vol. 12 No. 20 4835-4840 (2004)

Non-patent document 6: Sato & Kawakami, Optronics, July, 2001, page 197

Patent document 1: US 2003/0227415 A1

Patent document 2: US 2002/0175693 A1

BRIEF SUMMARY OF THE INVENTION

According to a first aspect of the present invention, there is provided a lens formed of a medium indicating negative refraction, having a convex or concave shaped smooth non-spherical surface or a non-circle curve.

According to a second aspect of the present invention, there is provided a lens having two optical surfaces formed of a medium indicating negative refraction, one of which is a rotational paraboloid, a rotational hyperboloid, a parabola, or a hyperbolic curve.

According to a third aspect of the present invention, there is provided a lens formed of a medium indicating negative refraction, having a convex shaped rotational paraboloid or rotational paraboloid, or alternatively, a convex shaped parabola or hyperbolic curve.

According to a fourth aspect of the present invention, there is provided a lens formed of a medium indicating negative refraction, having a convex shaped rotational paraboloid or rotational hyperboloid, or alternatively, a convex shaped parabola or hyperbolic curve and a flat surface.

According to a fifth aspect of the present invention, there is provided a lens having two optical surfaces formed of a medium indicating negative refraction, one of which is a concave shaped rotational paraboloid or rotational hyperboloid, or alternatively a parabola or hyperbolic curve.

According to a sixth aspect of the present invention, there is provided a lens formed of a medium indicating negative refraction, having a double concave shape, each of the double concave shape constituting a rotational paraboloid, the two rotational paraboloids sharing a common axis whose convex surfaces are opposed to each other.

According to a seventh aspect of the present invention, there is provided a lens having a double concave shape formed of a medium indicating negative refraction and having two spherical surfaces whose concave surfaces are opposite to each other, wherein, when respective curvature radiuses of the spherical surfaces are R₁ and R₂, a gap between the two surfaces satisfies the following formula 0.5(f ₁ +f ₂)<d<2(f ₁ +f ₂)  Formula 165-6

wherein f₁ is defined as a distance between P₁ and V₁, and f₂ is defined as a distance between P₂ and V₂, wherein P₁ is a focal point of a paraboloid of the curved surface S₁ and P₂ is a focal point of a paraboloid of the curved surface S₂, and V₁ is a cross point between the Z axis and the curved surface S₁, and V₂ is a cross point between the Z axis and the curved surface S₂.

According to an eighth aspect of the present invention, there is provided a lens formed of a medium indicating negative refraction, the lens having a double concave shape, wherein a rotational hyperboloid and a spherical surface are arranged such that respective convex surfaces thereof are opposed to each other.

According to a ninth aspect of the present invention, there is provided a lens formed of a medium indicating negative refraction, having a convex shaped rotational paraboloid or rotational hyperboloid, or alternatively, a convex shaped parabola or parabolic curve.

According to a tenth aspect of the present invention, there is provided a lens formed of a medium indicating negative refraction, one surface of the lens having a convex shaped rotational hyperboloid or a convex shaped hyperbolic curve, and the other surface of the lens having a flat surface.

According to an eleventh aspect of the present invention, there is provided an optical system having two lenses each of which is formed of a medium indicating negative refraction, which have a convex shaped rotational paraboloid or a convex shaped parabolic curve, wherein a convex surface of the rotational paraboloid or the parabola are opposed to each other.

According to a twelfth aspect of the present invention, there is provided an optical system having two lenses each of which is formed of a medium indicating negative refraction, having a convex shaped rotational paraboloid or a convex shaped parabolic curve at one surface and a flat surface at the other surface, wherein convex surfaces of the rotational paraboloids or the parabolas of the two lenses are opposed to each other.

According to a thirteenth aspect of the present invention, there is provided an optical system having an optical element made of a medium indicating negative refraction, wherein the optical system includes a spherical surface shaped optical surface and satisfies the following formulas

$\begin{matrix} {{{- 30}\lambda} \leqq {\delta{\int_{E}^{G}{{n(r)}\ {\mathbb{d}s}}}} \leqq {30\lambda\mspace{14mu}{and}}} & {{Formula}\mspace{14mu} 200\text{-}3} \\ {{{{- 30}\lambda} \leqq {\int_{E}^{G}{{n(r)}\ {\mathbb{d}s}}} \leqq {30\lambda}}\mspace{14mu}} & {{Formula}\mspace{14mu} 203\text{-}3} \end{matrix}$

wherein λ denotes a wavelength of a light beam to be used, δ represents a variation of an optical path, n(r) represents a refractive index in a positional vector r on the optical path, and “ds” represents a line element along the optical path.

According to a fourteenth aspect of the present invention, there is provided an optical system including a medium indicating negative refraction and a medium having a positive refractive index, wherein the optical system includes a spherical surface shaped optical surface and satisfies the following formulas

$\begin{matrix} {{{- 30}\lambda} \leqq {\delta{\int_{E}^{G}{{n(r)}\ {\mathbb{d}s}}}} \leqq {30\lambda\mspace{14mu}{and}}} & {{Formula}\mspace{14mu} 200\text{-}3} \\ {{{{- 30}\lambda} \leqq {\int_{E}^{G}{{n(r)}\ {\mathbb{d}s}}} \leqq {30\lambda}}\mspace{14mu}} & {{Formula}\mspace{14mu} 203\text{-}3} \end{matrix}$

wherein λ denotes a wavelength of a light beam to be used, δ represents a variation of an optical path, n(r) represents a refractive index in a positional vector r on the optical path, and “ds” represents a line element along the optical path.

Advantages of the invention will be set forth in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Advantages of the invention may be realized and obtained by means of the instrumentalities and combinations particularly pointed out hereinafter.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING

The accompanying drawings, which are incorporated in and constitute a part of the specification, illustrate embodiments of the invention, and together with the general description given above and the detailed description of the embodiments given below, serve to explain the principles of the invention.

FIG. 1 is a view for explaining a first embodiment of the present invention;

FIG. 2 is a view showing a modified example of the first embodiment;

FIG. 3 is a view for explaining a second embodiment of the present invention;

FIG. 4 is a view for explaining a third embodiment of the present invention;

FIG. 5 is a view for explaining a fourth embodiment of the present invention;

FIG. 6 is a view for explaining an emitted light component at the time when a light beam emitted from an object point E is formed as an image at an image point G;

FIG. 7 is a view showing an optical system composed of N faces;

FIG. 8 is a view showing a first specific example of a photonic crystal 340;

FIG. 9 is a view showing a second specific example of the photonic crystal 340; and

FIG. 10 is a view for explaining image formation using a parallel flat plate 380 formed of a negative refractive index medium 301.

DETAILED DESCRIPTION OF THE INVENTION

Hereinafter, preferred embodiments of the present invention will be described in detail with reference to the accompanying drawings.

FIG. 1 is a view for explaining a first embodiment of the present invention. A lens 410 is formed of a negative refractive index medium 301 comprising curved surfaces S₁ and S₂ as two optical surfaces. In the case of three dimensions, S₁ and S₂ are formed in the shape of a rotational paraboloid with a Z axis being a rotary axis. These curved surfaces are arranged such that convex surfaces sharing two axes are opposed to each other. In the case of two dimensions, S₁ and S₂ are provided as parabolas with the Z axis being an axis.

Hereinafter, although a description will be given with respect to the case of three dimensions, this description is also applicable to the case of two dimensions. In FIG. 1, a positive direction of an x axis is provided as a direction oriented from top to bottom of paper. A symbol designated by reference numeral 100 represents this direction. A focal point of a paraboloid of the curved surface S₁ is defined as P₁. A focal point of a paraboloid of the curved surface S₂ is defined as P₂. P₁ and P₂ are present on a Z axis. The Z axis is defined as an optical axis. A cross point between the Z axis and the surface S₁ is defined as V₁. A cross point between the Z axis and the surface S₂ is defined as V₂. A distance between P₁ and V₁ is defined as f₁. A distance between P₂ and V₂ is defined as f₂. At this time, f₁>0 and f₂>0 are established.

A reference surface is represented by K (in the case of two dimensions, K is a base line.) A cross point between K and the Z axis is defined as Q. K is orthogonal to the Z axis at Q. Then, it is better to select a distance d between V₁ and V₂ such that a distance between V₁ and Q is equal to f₁ and a distance between V₂ and Q is equal to f₂. d denotes a thickness of the lens 410.

Let us consider a light beam C emitted from P₁, the beam forming an angle θ with respect to the Z axis. A cross point between C and S₁ is defined as U₁. A cross point between C and S₂ is defined as U₂. An incident angle of C at the point U₁ is defined as i₁. A refraction angle of C at the point U₁ is defined as r₁. An incident angle of C at the point U₂ is defined as i₂. A refraction angle of C at the point U₂ is defined as r₂.

A refractive index of the negative refractive index medium 301 with respect to a vacuum is defined as n_(s). A refractive index of the medium at the periphery of the lens 410 with respect to a vacuum is defined as n₀. A cross point between C and K is defined as Y.

Now, in FIG. 1, a case in which the following formula is established is considered: n _(s) =−n ₀  Formula 104

In accordance with the refraction law, the following formula is established: n _(s) sin r ₁ =n ₀ sin i ₁  Formula 160

Thus, using Formula 104, the following formula is obtained: r ₁ =−i ₁  Formula 161

Then, the surface S₁ is provided as a rotational paraboloid with P₁ being a focal point, and thus, the light beam C after refracted becomes parallel to the Z axis on the surface S₁.

Namely, the line segments U₁ and U₂ are parallel to the Z axis (Hereinafter, in the case where two symbols representing points are referred together, these symbols are assumed to represent a line segment connecting two points with each other, and the term “line segment” is not omitted.).

With respect to refraction at the point U₂, the following formula is established similarly: r ₂ =−i ₂  Formula 162

Therefore, at S₂, C after refracted passes through P₂ that is a focal point of the surface S₂.

The above formula is established with respect to an arbitrary angle θ. Accordingly, assuming that an object point is placed at the point P₁, the emitted light components of all the light beams emitted from the object point are formed as an image at P₂ with no aberration.

At this time, an image forming magnification β is given by the following formula:

$\begin{matrix} {\beta = {- \frac{f_{2}}{f_{1}}}} & {{Formula}\mspace{14mu} 163} \end{matrix}$

Since f₁ and f₂ can be arbitrarily selected by changing the shape of surfaces S₁ and S₂, an image forming optical system with no aberration, the optical system having an arbitrary magnification, can be obtained by using the lens 410.

It is better to establish the following formula because a magnified image or a reduced image can be obtained: |β|≠1  Formula 163-2

It is still better to establish the following formula because a large magnified image can be obtained: |β|>2  Formula 163-3

It is further better to establish the following formula: |β|>5  Formula 163-4

It is still further better to establish the following formula because a reduced image can be obtained: |β|>½  Formula 163-5

It is better to establish the following formula: |β|<⅕  Formula 163-6

Of course, it is allowed to establish the following formula: |β|=1  Formula 163-7

The system shown in FIG. 1 can be used for an objective lens of a microscope, a stepper, an optical disk, a television camera, a digital camera, an optical LSI and the like.

As an embodiment, the following settings are provided:

$\begin{matrix} {f_{1} = {1\mspace{14mu}{mm}}} \\ {f_{2} = {5\mspace{14mu}{mm}}} \\ {d = {6\mspace{14mu}{mm}}} \\ {n_{0} = {1\mspace{14mu}({vacuum})}} \\ {n_{s} = {- 1}} \\ {{NA}\mspace{14mu}\left( {{incident}\mspace{14mu}{side}\mspace{14mu}{numerical}\mspace{14mu}{aperture}\mspace{14mu} 0.98} \right)} \\ {{{MWD}_{1} = {0.2\mspace{14mu}{mm}}},\mspace{14mu}{{MWD}_{2} = {4\mspace{14mu}{mm}}}} \\ {\beta = {- 5}} \\ {\lambda = {488\mspace{14mu}{nm}\mspace{14mu}\left( {\lambda\mspace{14mu}{denotes}\mspace{14mu}{an}\mspace{14mu}{optical}\mspace{14mu}{wavelength}} \right)}} \end{matrix}$

Here, MWD denotes a mechanical working distance. A distance obtained by measuring a length between P₁ and a portion A that is the closest to P₁ of the lens 410 in parallel to the Z axis is defined as MWD₁. If the portion A is present at the right side from P₁, a flat surface on which an object is present can be observed without collision with the lens 410. Therefore, it is desirable that the following formula is established: MWD₁>0  Formula 164

It is still better to establish the following formula: MWD₁>0.1 mm  Formula 165

MWD₂ denotes a distance obtained by measuring a length between P₂ and a portion B that is the closest to P₂ of the lens 410 in parallel to the Z axis. In order to avoid colliding between an image flat surface and the lens 410, it is desirable that the following formula is established: MWD₂>0  Formula 164-2

It is still better to establish the following formula: MWD₂>0.1 mm  Formula 165-2

The following formula is also established, but a slight error is allowed: d=f ₁ +f ₂  Formula 165-3

Image formation with high precision can be achieved when the following formula is established: 0.9(f ₁ +f ₂)<d<1.1(f ₁ +f ₂)  Formula 165-4

Practically sufficient image formation can be achieved when the following formula is established: 0.7(f ₁ +f ₂)<d<1.3(f ₁ +f ₂)  Formula 165-5

Depending on usage, the following formula will suffice: 0.5(f ₁ +f ₂)<d<2(f ₁ +f ₂)  Formula 165-6

Now, let us consider an optical path length of a light beam emitted from an object point placed at P₁.

From a definition of a paraboloid or a parabola, the following formulas are established: P ₁ U ₁ =U ₁ Y  Formula 166 YU ₂ =U ₂ P ₂  Formula 167

Therefore, the following formula is obtained: P ₁ U ₁ +U ₂ P ₂ =U ₁ Y+YU ₂  Formula 168

In a positive refractive index medium, an evanescent wave increases in an exponential manner. In a negative refractive index medium, an evanescent wave decreases in an exponential manner.

The right side of Formula 168 is an optical path length in a positive refractive index medium while the left side of Formula 168 is an optical path length in a negative refractive index medium. Since both of these sides are equal to each other and Formula 104 is established, a sum of all the optical path lengths is 0. It is better to establish the following formula:

$\begin{matrix} {0.8 < {\frac{n_{s}}{n_{0}}} < 1.2} & {{Formula}\mspace{14mu} 169} \end{matrix}$

This is because, as long as a value of |n_(s)/n₀| is within the above range, the lowering of image formation performance is less. Depending on usage of an optical system, it is better to establish the following formula:

$\begin{matrix} {0.5 < {\frac{n_{s}}{n_{0}}} < 1.6} & {{Formula}\mspace{14mu} 169\text{-}2} \end{matrix}$

For the purpose of image formation with no aberration for radiation completion, Formula 165-3, Formula 165-4, and Formula 165-5 may not be satisfied.

In an example shown in FIG. 1, a refractive index may be inverted. Namely, the left side space with respect to S₁ may be filled with a negative refractive index medium, the right side space from S₂ may be filled with a negative refractive index medium, and a space between S₁ and S₂ may be filled with a positive refractive index medium. As long as Formula 104 is established, image formation with no aberration is achieved.

Alternatively, a space at the left side with respect to S₁ and at the right side from a flat surface passing through P₁ and vertical to the Z axis, may be filled with a negative refractive index medium to produce one lens; a space at the right side from S₂ and being at the right side from a flat surface passing through P₂ and vertical to the Z axis may be filled with a negative refractive index medium to produce another lens; and a space between S₁ and S₂ may be filled with a positive refractive index medium. As long as Formula 104 is established, image formation with no aberration is achieved. This appearance is shown in FIG. 2.

Although in the lens 410 shown in FIG. 1, the shape of the surfaces S₁ and S₂ has been a rotational paraboloid, this shape may be approximated by a spherical surface.

When a curvature radius of the surface S₁ is defined as R₁, and a curvature radius of the surface S₂ is defined as R₂, R₁<0 and R₂>0 are established, and the following formula is established:

$\begin{matrix} {f_{1} = {\frac{R_{1}}{2}}} & {{Formula}\mspace{14mu} 170} \\ {f_{2} = {\frac{R_{2}}{2}}} & {{Formula}\mspace{14mu} 171} \end{matrix}$

It is still better to establish the following formula: d=1/2(|R ₁ +|R ₂|)  Formula 172

In the case of two dimensions, a circle may be placed instead of a spherical surface. Formula 165-3, Formula 165-4, Formula 165-5, and Formula 165-6 are established in both of the cases of a circle and a spherical surface.

In the example shown in FIG. 1, an image pickup device is produced by placing an image pickup element 408-2 at a position P₂. This device is featured in that a magnified image or a reduced image can be obtained with no aberration. A microscope capable of television observation, a digital camera, a television camera or the like can be obtained.

In addition, when a light source such as a lamp or an LED is placed at a position P₁, an illumination optical system for focusing light at a position P₂ can be obtained. The system can be used for a light source for a microscope, an endoscope or the like.

Further, in the example shown in FIG. 1, a photo mask is placed at P₂, a wafer is placed at P₁, and then, illumination light is emitted from the right side, so that a stepper (projection exposure device) for use in manufacture of an LSI or the like can be obtained. A stepper optical system is also called a lithography optical system.

FIG. 3 is a view for explaining a second embodiment of the present invention, and shows a microscope 422 using a lens 410. The light beams emitted from a light source 303 are focused by an illumination lens 423, and illuminate a sample 307. The light beams scattered from the sample 307 are formed as an image by the lens 410 having a function as an objective lens, an optical path thereof is bent on a mirror 424, and an actual image is produced on an I.P. The sample 307 corresponds to P₁, and the I.P. corresponds to P₂. Then, the image is magnified by an ocular lens 308, and a magnified image can be seen by an eye 309.

Alternatively, a magnified image can be picked up by a television camera 425 provided rearward of the ocular lens 308. Here, reference numeral 426 denotes an image pickup lens, 408 denotes a solid state image pickup element, 427 denotes an electronic circuit, and 428 denotes a display device.

The ocular lens 308 and the image pickup lens 426 are made of a medium having a positive refractive index, such as a glass or a plastic.

The microscope 422 is placed in air of 1 atmosphere, and the settings are provided as follows:

$\begin{matrix} {n_{s} = {- 1.0003}} \\ {n_{0} = 1.0003} \\ {\lambda = {500\mspace{14mu}{nm}}} \\ {{\beta } = 20} \end{matrix}$

In an optical system shown in FIG. 3, an optical disk 323 is arranged instead of the sample 307, a translucent mirror 305 is arranged instead of the mirror 424, a photo detector 324 is arranged at a position of IP, and further, a light source such as a semiconductor laser is arranged at a position of the light source 321, whereby an optical disk or an optical device can be obtained. This device is featured in that large NA can be obtained with no aberration.

FIG. 4 is a view for explaining a third embodiment of the present invention. In the figure, an illumination optical system 413 uses a lens 412 formed of a negative refractive index medium 301. A left side face 413 of the lens 412 is provided as a rotational paraboloid, and a right side face 414 of the lens 412 is provided as a flat surface. An axis of the illumination optical system 413 is provided as a Z axis.

A light source 415 is arranged at a focal point P₁ of the illumination optical system 413, and the light beams emitted from the light source 415 are refracted by the illumination optical system 413, and are emitted as light beams parallel to the Z axis. In this optical system, a magnification β is infinitely large. Then, image formation with no aberration can be carried out in an infinitely telescopic manner.

As compared with a conventional illumination system using a reflection cover, a light source itself does not block an optical path, and an efficient illumination optical system can be achieved.

It is easy to set NA at the light incident side to be 1 or more, namely, θ≧90°. The shape of the surface 414 may be a curved surface, a fresnel surface or the like. Examples of light sources include a halogen lamp, a xenon lamp, an LED, a semiconductor laser, an emission end of an optical fiber, an optical waveguide, and a super-luminescent diode. In an example of the illumination optical system 413, the periphery of the lens 412 is air, and n_(s)=−1.0003, n₀=1.003, θ=110°, P₁=10 mm, and λ (wavelength) ranges from 350 to 700 nm.

The shape of the illumination optical system 413 may be approximated on a spherical surface. In that case, assuming that the curvature radius is R₁, the following formula is established:

$\begin{matrix} {f_{1} = {\frac{R_{1}}{2}}} & {{Formula}\mspace{14mu} 170} \end{matrix}$

In the case of two dimensions, the shape of the illumination optical system 413 becomes a parabola or a circle.

In an example shown in FIG. 4, Formula 164 may be satisfied. An illumination system shown in FIG. 4 can be used, for example, for the purpose of display by combining it with an LED, and a search light, a light source of a microscope, a light source of an endoscope, a light source of optical communication, and a light source of an optical disk by combining it with a lamp.

In the case where a distance by which light emission is attempted by the illumination optical system is a finite distance, the shape of the illumination optical system 413 may be a rotational hyperboloid or a hyperbolic curve (in the case of two-dimension) with the Z axis being a long axis. In this case as well, the position of the light source is a focal point of one of a hyperboloid and a hyperbolic curve. A lens made of a negative refractive index medium having a hyperboloid will be described later with reference to FIG. 5.

In the optical system shown in FIG. 4, conversely, light beams are incident from the right side, and an image pickup element 408-2 is arranged at a position of P₁, whereby an image pickup device with no aberration can be obtained. An image pickup device having a large numerical aperture at the emission side and sufficient brightness can be obtained.

This device can be used for a digital camera, a television camera or the like. Alternatively, in the optical system shown in FIG. 4, the light source is removed from the position of P₁, an ocular lens 308 is placed at the left side of P₁, and light beams are incident from the right side, so that a telescope can be obtained. Formula 164, Formula 165, Formula 169, and Formula 169-2 may be applied to the system shown in FIG. 4. Depending on usage, the shape of the surface 414 may be a curved surface.

FIG. 5 is a view for explaining a fourth embodiment of the present invention. In the figure, an optical system 452 includes a lens 450 having a rotational hyperboloid S₁ made of a negative refractive index medium 301. A lens 451 has a spherical surface made of the negative refractive index medium 301.

P₁ and P₂ denote as focal points of the rotational hyperboloid (hyperbolic curve in the case of two dimensions). A rotation axis (in the case of two dimensions, an axis) of S₁ is a X axis. S₁ denote a left side face of the lens 450, S₂ denotes a right side face of the lens 450, S₃ denotes a left side face of the lens 451, and S₄ denotes a right side face of the lens 451.

V_(i)(i=1, 2, 3, 4) denotes a cross point between S₁ and the Z axis. U₁ denotes a cross point between a light beam C emitted from P₁ and S₁. S₂, S₃, and S₄ denote a spherical surface (circle in the case of two dimensions) with P₂ being a spherical center, and its radius is defined as R_(i). In this case, the following formula is established: R _(i)>0 (i=2, 3, 4)  Formula 180

d_(i) denotes a distance between V_(i) and V_(i+1), d₄ denotes a distance between V₄ an P₂. g denotes a distance between P₁ and V₁.

Here, d₁ and d₃ are selected so as to satisfy the following formulas:

$\begin{matrix} {d_{1} = g} & {{Formula}\mspace{14mu} 181} \\ \begin{matrix} {d_{3} = {d_{2} + d_{4}}} \\ {= {{1/2}\;\left( {{P_{1}P_{2}} - 2_{g}} \right)}} \end{matrix} & {{Formula}\mspace{14mu} 182} \end{matrix}$

A refractive index of the lens 450 with respect to a vacuum is defined as n_(s), a refractive index of the lens 451 with respect to a vacuum is defined as n₂, and a refractive index of the periphery of the lenses 450 and 451 is defined as n₀.

n_(s) and n₂ are selected so as to satisfy the following formulas:

$\begin{matrix} \begin{matrix} {n_{s} = {- n_{0}}} \\ {n_{2} = {- n_{0}}} \end{matrix} & {{Formula}\mspace{14mu} 184} \end{matrix}$

Therefore, the following formula is established: n₂=n_(s)  Formula 185

Definitions of i₁ and r₁ are an incident angle and a refraction angle at U₁, respectively.

From Formula 181, the following formula is obtained: R ₂ =P ₁ P ₂−2g  Formula 185-2

Assume that S₂, S₃, and S₄ are not present. r₁=−i₁ is established in accordance with the refraction law, and C passes through P₂ independent of θ in accordance with geometrical property of the rotational hyperboloid and hyperbolic curve.

Even if S₂, S₃, and S₄ are present, these surfaces are concentric surfaces with P₂ being a spherical center, and thus, U₂, U₃, and U₄ are orthogonal to the respective surfaces, no refraction occurs, and C passes through P₂. For this reason, whatever the lens 451 is present or absent, the light beams emitted from P₁ by means of the lens 450 is formed as an image at P₂ with no aberration. Therefore, the emitted light components of the light beams emitted from P₁ get together at P₂ In other words, an actual image of P₁ is formed at P₂. At this time, a magnification β is given by the following formula:

$\beta = {- \frac{g}{{P_{1}P_{2}} - g}}$

Next, let us consider an optical path length of light beams emitted from P₁. In addition, let us consider a case in which the lens 451 is present. By definitions of rotational hyperboloids (hyperbolic curves in two dimensions), the following formula is established: P ₁ U ₁ =U ₁ U ₂  Formula 186

In addition, the following formula is also established:

$\begin{matrix} \begin{matrix} {{U_{3}U_{4}} = {{U_{2}U_{3}} + {U_{4}P_{2}}}} \\ {= d_{3}} \\ {= {d_{2} + d_{4}}} \end{matrix} & {{Formula}\mspace{14mu} 187} \end{matrix}$

Thus, the optical path length in a positive refractive index medium of the light beam C is equal to that in a negative refractive index medium. In practice, instead of Formula 181, the following formula may be established: 0.7g≦d ₁≦1.4g  Formula 181-2

Depending on usage, the following formula is also allowed: 0.3g≦d ₁≦2.5g  Formula 181-3

In practice, instead of Formula 185-2, the following formula may be established: 0.7(P ₁ P ₂−2g)≦R ₂≦1.5(P ₁ P ₂−2g)  Formula 185-2-1

Depending on usage, the following formula is also allowed: 0.3(P ₁ P ₂−2g)≦R ₂≦3(P ₁ P ₂−2g)  Formula 185-2-2

In practice, instead of Formula 182, the following formula may be established: 0.35(P ₁ P ₂−2g)≦d ₃≦0.8(P ₁ P ₂−2g)  Formula 182-2

Depending on usage, the following formula is also allowed: 0.15(P ₁ P ₂−2g)≦d ₃≦1.6(P ₁ P ₂−2g)  Formula 182-3

In addition, the lens 451 may be divided into a plurality of concentric spherical surfaces. Further, a sum of their thickness may be equal to the following formula:

$\frac{1}{2}\left( {{P_{1}P_{2}} - {2g}} \right)$

As a special case, d₂=0 may be defined, and d₃=d₄ may be defined. Namely, the lens 450 and the lens 451 are integrated with each other. At this time, the following formula is established:

$\begin{matrix} {{V_{1}V_{4}} = {\frac{1}{2}P_{1}P_{2}}} & {{Formula}\mspace{14mu} 188} \end{matrix}$

In this case, V₁V₄ represents a thickness of the lens 450. In practice, the following formula may be established: 0.35P ₁ P ₂ ≦V ₁ V ₄≦0.8P ₁ P ₂  Formula 188-2

Depending on usage, the following formula may be established: 0.15P ₁ P ₂ ≦V ₁ V ₄≦1.5P ₁ P ₂  Formula 188-3

The usage of the optical system shown in FIG. 5 is identical to those of the examples shown in 1 to 4.

In addition, an object is placed at P₂ so that an image may be formed at P₁.

The surface S₁ may be approximated on a spherical surface of a radius of −2 g.

Formula 163-2, Formula 163-3, Formula 163-4, Formula 163-5, Formula 163-6, Formula 164, Formula 165, Formula 164-2, and Formula 165-2 can be applied to the example shown in FIG. 5.

Formula 169 and Formula 169-2 can be applied for the lens 450, and can also be applied for the lens 451 if n_(s) is replaced with n₂.

Formula 173 and Formula 174 described later can be applied if f₁ is replaced with g.

In an embodiment shown in FIG. 5, the following settings are provided:

$\begin{matrix} {g = {100\mspace{14mu}{µm}\mspace{14mu}({micrometers})}} \\ {{P_{1}P_{2}} = {500\mspace{14mu}{µm}}} \\ {d_{2} = {30\mspace{14mu}{µm}}} \\ {d_{3} = {150\mspace{14mu}{µm}}} \\ {d_{4} = {120\mspace{14mu}{µm}}} \\ {R_{2} = {300\mspace{14mu}{µm}}} \\ {\theta = {80{^\circ}}} \\ {n_{s} = {{- 1.}\; 00028}} \\ {n_{2} = {- 1.00028}} \\ {n_{0} = 1.00028} \\ {\lambda = {1.5\mspace{14mu}{µm}}} \end{matrix}$

The optical system shown in FIG. 5 can be used for an optical pickup, an optical LSI, a microscope, a lithography optical system, and the like.

A description will be given with respect to an allowable value Δ of a deviation in the Z direction from P₁, of an object point or an object, or alternatively, a light source.

It is sufficient to establish the following formula for the purpose of utilizing a general optical device:

$\begin{matrix} {\Delta \leqq \frac{f_{1}}{7}} & {{Formula}\mspace{14mu} 173} \end{matrix}$

Depending on usage, the following formula can also be allowed:

$\begin{matrix} {\Delta \leqq \frac{f_{1}}{3}} & {{Formula}\mspace{14mu} 174} \end{matrix}$

In the case of a rotational hyperboloid or hyperbolic curve, it is assumed that f₁ is replaced with g.

A refractive index may be inverted in the system shown in FIG. 5. Namely, a left side space from S₁, a space between S₂ and S₃, and a right side space from S₄ may be filled with a negative refractive index medium, and a space between S₁ and S₂ and a space between S₃ and S₄ may be filled with a positive refractive index medium. As long as Formula 104 is satisfied, image formation with no aberration is achieved.

Further, a space being at the right side from a flat surface passing through P₁ and vertical to the Z axis, and being at the left side from S₁, may be filled with a negative refractive index, and may be used as one lens. At this time, the space at the right side of S₁ and at the left side of S₂ is defined as a positive refractive index medium. As long as Formula 104 is satisfied, image formation with no aberration is achieved.

Referring now to FIG. 6, when the light beams emitted from an object point E are generally formed as an image at an image point G, it is necessary to establish the following formula with respect to the emitted light components:

$\begin{matrix} {{\delta{\int_{E}^{G}{{n(r)}\ {\mathbb{d}s}}}} = {0\mspace{14mu} d}} & {{Formula}\mspace{14mu} 200} \end{matrix}$

This is a Felmat's minimum time theory. In Formula 200, n(r) represents a refractive index in a positional vector r on an optical path.

“ds” represents a line element along an optical path.

Assuming that the following formula is established:

$\begin{matrix} {r = \begin{pmatrix} x \\ y \\ z \end{pmatrix}} & {{Formula}\mspace{14mu} 201} \end{matrix}$

the following formula is also established: ds=√{square root over ((dx)²+(dy)²+(dz)²)}{square root over ((dx)²+(dy)²+(dz)²)}{square root over ((dx)²+(dy)²+(dz)²)}  Formula 202

δ preceding an integral symbol represents a variation of an optical path. On the other hand, with respect to evanescent wave components, it is desirable to establish the following formula relevant to the light beams emitted in a vertical direction of an object surface:

$\begin{matrix} {{{\int_{E}^{G}{{n(r)}\ {\mathbb{d}s}}} = 0}\mspace{11mu}} & {{Formula}\mspace{14mu} 203} \end{matrix}$

This formula is satisfied on the Z axis in the examples shown in FIGS. 1, 5 and 10. In order to establish Formula 203, it is necessary that both of a negative refractive index medium and a positive refractive index medium exist on an optical path.

In practice, instead of Formula 200, the following formula may be established:

$\begin{matrix} {{{{- 10}\lambda} \leqq {\delta{\int_{E}^{G}{{n(r)}\ {\mathbb{d}s}}}} \leqq {10\lambda}}\mspace{14mu}} & {{Formula}\mspace{14mu} 202\text{-}2} \end{matrix}$

Depending on usage, the following formula is also allowed:

$\begin{matrix} {{{{- 30}\lambda} \leqq {\delta{\int_{E}^{G}{{n(r)}\ {\mathbb{d}s}}}} \leqq {30\lambda}}\mspace{14mu}} & {{Formula}\mspace{14mu} 200\text{-}3} \end{matrix}$

Similarly, instead of Formula 203, the following formula may be established:

$\begin{matrix} {{{{- 10}\lambda} \leqq {\int_{E}^{G}{{n(r)}\ {\mathbb{d}s}}} \leqq {10\lambda}}\mspace{14mu}} & {{Formula}\mspace{14mu} 203\text{-}2} \end{matrix}$

Depending on usage, the following formula is also allowed:

$\begin{matrix} {{{{- 30}\lambda} \leqq {\int_{E}^{G}{{n(r)}\ {\mathbb{d}s}}} \leqq {30\lambda}}\mspace{14mu}} & {{Formula}\mspace{14mu} 203\text{-}3} \end{matrix}$

In the formula, λ denotes a wavelength of a light beam to be used.

As shown in FIG. 7, let us consider an optical system composed of N faces. n₁ denotes a refractive index of a medium between an i face and an i+1 face. “i” denotes an integer from 0 to N. In order to establish better image formation in such an optical system, it is desirable that Formula 200 and Formula 203 are established.

It is advantageous to provide an optical system including three or more curved surfaces because aberration can be easily eliminated.

Further, it is advantageous to form a non-spherical surface in order to correct for aberrations. With respect to an image forming relationship, it is good to provide an optical system that does not relay an image because aberration is improved. The term “relay” used here denotes that once formed image is further formed as an image, and an optical system shown in FIG. 10 falls under this relay system. It is desirable that “n_(i)” includes a value equal to a value whose sign is reversed and whose absolute value is equal thereto. For example, n₀=1, n₁=−1, n₂=2, n₃=−2 or the like is desirable. Although this is commonly true for the present application, a medium having a uniform refractive index should be used as a negative refractive index substance because it is easily fabricated.

Formulas 200-2, 200-3, 203-2, and 203-3 are applied similarly to an example shown in FIG. 7 as well.

In the case where a circle, a parabola, a hyperbolic curve, and an ellipse are described in the present application, these shapes are assumed to include a cylindrical surface, a cylindrical paraboloid, a cylindrical hyperboloid, and a cylindrical elliptical surface as well. In addition, in the present application, a sign of a radius of a spherical surface or a circle is assumed to be positive in the case where the surface is formed in a convex shape on the left. Namely, the sign is assumed to be positive in the case where the surface is formed in a convex shape in the −Z axis direction.

A rotational paraboloid, a rotational hyperboloid, a rotational elliptical surface, a cylindrical paraboloid, a cylindrical hyperboloid and a cylindrical elliptical surface each are provided as an example of a smooth non-spherical surface.

A parabola, a hyperbolic curve, and an ellipse each are provided as a smooth curve that is not a circle.

A two-dimensional optical system or optical element according to the present invention may be used for an optical signal processor system such as an optical circuit, an optical IC, or an optical LSI.

In the case where the term “light beam” is used in the present invention, the term is assumed to include an electromagnetic wave having an arbitrary number of vibrations. In addition, when a wavelength is defined as λ, there is attained a practical merit in using an electromagnetic wave given by the following formula: 100 nm≦λ≦20 cm  Formula 280

The electromagnetic wave may be used for a ultraviolet-ray, a visible light beam, a near-infrared ray, an infrared ray, a far-infrared ray, a Terra-Herz wave, a microwave or the like. In order to mitigate an effect of color aberration that a negative refractive index medium has, a light beam having a single wavelength or a light beam having a narrow wavelength region (for example, having a width of 100 nm or less) may be used.

Now, the contents common to the present invention will be described here. A photonic crystal can be exemplified as a specific substance of a negative refractive index medium 301. FIG. 8 shows a first specific example of a photonic crystal 340, and FIG. 9 shows a second specific example of the photonic crystal 340. As shown in FIGS. 8 and 9, the photonic crystal 340 is provided as a substance having a cyclic structure in order of λ to 1λ/n, and is produced in accordance with lithography or the like. The materials include: dielectrics such as synthetic resins such as SiO₂, acryl, and polycarbonate, and GaAs. Here, λ denotes a wavelength of a light beam to be used. Values of repetition cycles Sx, Sy, and Sz in the X, Y, and Z directions in the figure each have a value in order of λ to 1λ/n. It is known that a negative refractive index can be achieved in the vicinity of a band end of the photonic crystal (refer to non-patent document 3). The z direction in the figure may be an optical axis of an optical system.

The Z axis denotes a direction of an axis that is the next in rotational symmetry of a photonic crystal.

It is desirable that Sx, Sy, and Sz satisfy any of the following formulas: λ/10<Sx<λ  Formula (5-1) λ/10<Sy<λ  Formula (5-2) λ/10<Sz<λ  Formula (5-3)

If the values of Sx, Sy, and Sz each exceed the upper limit or are lower than the lower limit, the photonic crystal does not function.

Depending on usage, it suffices that any of the following formulas may be satisfied: λ/30<Sx<4λ  Formula (5-4) λ/30<Sy<4λ  Formula (5-5) λ/30<Sz<4λ  Formula (5-6)

It is known that, with respect to a negative refractive index medium, a refractive index of the medium is obtained as follows when a specific dielectric ε of the medium is negative and a specific permeability μ of a medium is negative: −√{square root over (εμ)}

Usable examples of a negative refractive index medium include: a substance indicating negative refraction; a substance indicating approximately negative refraction, such as a thin film made of silver, gold, or copper; a substance indicating a negative refractive index with respect to a specific polarizing direction; and a thin film made of a substance whose dielectric ε is substantially −1.

A negative refractive index medium may be called a Left handed material. In the present invention, all the negative refractive index mediums, left handed materials, substances indicating approximately negative refraction, substances indicating a negative refractive index with respect to a specific polarizing direction, and a thin film of a substance whose dielectric ε is substantially −1 or the like are called substances indicating negative refraction. A substance indicating complete image formation is also included in a medium indicating negative refraction. In addition, in the case of a thin film made of a substance whose dielectric ε is substantially −1, the following formula may be satisfied: −1.2<ε<−0.8  Formula (5-7)

Depending on usage, the following formula may be established: −1.6<ε<−0.5  Formula (5-8)

While the embodiment has described that an example in which a single-color light beam is mainly used as a wavelength of a light beam to be used, a light source of continuous spectra, a white light source, a sum of a plurality of single color light beams, or a low coherent light source such as a super-luminescent diode may be used without being limited thereto.

A wavelength of 0.1 μm to 3 μm may be used because it can be transmitted even in air and a light source can be easily obtained. A visible wavelength is further desirable because it is available more easily. It is still further desirable to define a wavelength of 0.6 μm or less because image resolution is improved.

It is yet further desirable to define to 20 m or less a length measured along an optical axis of an optical system including a negative refractive index medium because an optical system and an optical device are easily fabricated.

In an example shown in the embodiment of FIG. 1 of the present application, a distance between an object point (such as P₁) or an image point (such as P₂) relevant to an image forming optical system (such as 410) and the image forming optical system is featured to be finite. Further, even if the object point and the image point are replaced with each other, an image forming relationship is satisfied. Such an optical system is also included in the present application.

Moreover, while the present application has used a term “complete image formation”, this term is assumed to include a case in which 100% complete image formation is not carried out, for example, a case in which 50% image resolution is improved. Namely, for example, this term is assumed to include a case in which image resolution is improved to some extent more remarkably than a normal refraction limit.

According to the present invention, there is provided an optical system capable of producing a magnified image or a reduced image of an object by using a negative refractive index medium.

Lastly, a description will be given with respect to definitions of technical terms used in the present embodiment.

An optical device denotes a device including an optical system or an optical element. The device may not function as an optical device solely. Namely, the device may be part of another device.

An optical device includes an image pickup device, an observation device, a display device, an illumination device, a signal processor device, an optical information processing unit, a projector device, a projection exposure device and the like.

Examples of the image pickup device include a film camera, a digital camera, a PDA digital camera, a robot's eye, a lens exchange type digital single-lens reflex camera, a television camera, a mobile image recording device, an electronic mobile image recording device, a cam coder, a VTR camera, a digital camera of a portable cellular phone, a television camera of a portable cellular phone, an electronic endoscope, a capsule endoscope, a car-mounted camera, a camera of an artificial satellite, a camera of a planetary probe, a camera of a astronomic probe, a camera of a monitor device, eyes of various sensors, a digital camera of a recording device, an artificial sight, a laser scanning type microscope, a projection exposure device, a stepper, an aligner, and an optical probe type microscope. Examples of electronic image pickup devices include a digital camera, a card type digital camera, a television camera, a VTR camera, a mobile image recorder camera, a digital camera of a portable cellular phone, a television camera of a portable cellular phone, a car-mounted camera, a camera of an artificial satellite, a camera of a planetary probe, a camera of an astronomic probe, and a digital camera of a recording device.

Examples of the observation device include a microscope, a telescope, a glass, a binocular glass, a loupe, a fiber scope, a finder, a view finder, a contact lens, an intraocular lens, and an artificial sight.

Examples of the display device include a liquid crystal display, a view finder, a game machine (PlayStation available from Sony Corporation.), a video projector, a liquid crystal projector, a head mounted display (HMD), a personal digital assistant (PDA), a portable cellular phone, and an artificial sight.

A video projector, a liquid crystal projector and the like can be used as a projector device.

Examples of the illumination device include a camera strobe, an automobile headlight, an endoscopic light source, and a microscopic light source.

Examples of the signal processor device include: a portable cellular phone, a personal computer, a game machine, an optical disk read/write device, an optical calculator computing device, an optical interconnection device, an optical information processing unit, an optical LSI, an optical computer, and a PDA.

An information transmitter device designates a portable cellular phone, a fixed type telephone, a game machine, remote controllers of a television, a radio cassette, a stereo and the like; and devices capable of inputting and transmitting any kind of information contained in a personal computer, a keyboard, a mouse, a touch panel, etc. of a personal computer.

This transmitter device also includes a television monitor, a personal computer monitor, and a display with an image pickup device.

The information transmitter device is included in the signal processor device.

An image pickup element designates, for example, a CCD, an image pickup tube, a solid state image pickup element, a photographic film or the like. In addition, a parallel planer plate is assumed to be included in one of prisms. A change in degree of vision is assumed to be included in an observer's change. Object changes are assumed to include a change in distance relevant to an object, an object movement, an object motion, a vibration, an object blurring and the like. An image pickup element, a wafer, an optical disk, a silver salt film and the like are examples of image formation members.

Definitions of extended curved surfaces are as follows.

In addition to a spherical surface, a flat surface, and a rotational symmetrical non-spherical surface, there may be any shape of a spherical surface that is eccentrically deformed with respect to an optical axis, a flat surface, a rotational symmetrical non-spherical surface, or alternatively, a non-spherical surface having symmetrical surfaces, a non-spherical surface having only one symmetrical surface, a non-spherical surface free of a symmetrical surface, a free curved surface, a surface having a point and a line that cannot be differentiated. A reflection surface or a refraction surface may be used as long as it has any effect on a light beam.

In the present invention, these elements are generally called an extended curved surface.

An optical system includes an optical system and an image forming optical system or the like.

An image forming optical system designates an image pickup optical system, an observation optical system, a projection optical system, a projection and exposure optical system, a display optical system, a signal processing optical system or the like.

Examples of the image pickup optical system include an image pickup lens of a digital camera.

Examples of the observation optical system include a microscopic optical system and a telescopic optical system.

Examples of the projection optical system include a video projector optical system, a lithography optical system, an optical system for reading out or writing an optical disk, and an optical system of an optical pickup.

Examples of the projection and exposure optical system include a lithography optical system.

Examples of the display optical system include an optical system of a view finder of a video camera.

Examples of the signal processing optical system include an optical system for reading out and writing an optical disk, and an optical system of an optical pickup.

Examples of the signal processing optical system include.

Examples of the illumination optical system include optical systems used for a microscope, an endoscope, an optical pickup, a searchlight, and a lighthouse. Additional examples of the illumination optical system include a focusing lens system for a light emitting diode, a light source optical system for optical communication, and an optical system for optical LSI or the like.

Optical elements designate a lens, a non-spherical surface lens, a mirror, a prism, a free curved surface prism, a diffraction optical element (DOE), a non-uniform lens or the like. A parallel flat plate is included in one of the optical elements. 

1. A lens formed of a medium indicating negative refraction, having a convex or concave shaped smooth non-spherical surface or a non-circle curve.
 2. A lens having two optical surfaces formed of a medium indicating negative refraction, one of which is a rotational paraboloid, a rotational hyperboloid, a parabola, or a hyperbolic curve.
 3. A lens having two optical surfaces formed of a medium indicating negative refraction, one of which is a concave shaped rotational paraboloid or rotational hyperboloid, or alternatively a parabola or hyperbolic curve.
 4. A lens formed of a medium indicating negative refraction, having a double concave shape, each of the double concave shape constituting a rotational paraboloid, the two rotational paraboloids sharing a common axis whose convex surfaces are opposed to each other.
 5. A lens according to claim 4, wherein, when respective focal distances of the rotational paraboloid is defined as f₁ and f₂, a gap between the two surfaces satisfies the following formula 0.5(f ₁ +f ₂)<d<2(f ₁ +f ₂) wherein f₁ is defined as a distance between P₁ and V₁, and f₂ is defined as a distance between P₂ and V₂, wherein P₁ is a focal point of a paraboloid of the curved surface S1 and P₂ is a focal point of a paraboloid of the curved surface S₂, and V₁ is a cross point between the Z axis and the curved surface S₁, and V₂ is a cross point between the Z axis and the curved surface S₂.
 6. A lens according to claim 5, wherein a parabolic curve is used instead of the rotational paraboloid.
 7. A lens having a double concave shape formed of a medium indicating negative refraction and having two spherical surfaces whose concave surfaces are opposite to each other, wherein, when respective curvature radiuses of the spherical surfaces are R₁ and R₂, a gap between the two surfaces satisfies the following formula 0.5(f ₁ +f ₂)<d<2(f ₁ +f ₂) wherein f₁ is defined as a distance between P₁ and V₁, and f₂ is defined as a distance between P₂ and V₂, wherein P₁ is a focal point of a paraboloid of the curved surface S₁ and P₂ is a focal point of a paraboloid of the curved surface S₂, and V₁ is a cross point between the Z axis and the curved surface S₁, and V₂ is a cross point between the Z axis and the curved surface S₂.
 8. A lens according to claim 7, wherein a circle is used instead of the spherical surface.
 9. A lens formed of a medium indicating negative refraction, the lens having a double concave shape, wherein a rotational hyperboloid and a spherical surface are arranged such that respective convex surfaces thereof are opposed to each other.
 10. A lens according to claim 9, wherein a spherical center of the spherical surface is present on a rotary axis of the rotational hyperboloid.
 11. A lens according to claim 9, wherein the lens satisfies either one of the following formulas 0.15P ₁ P ₂ ≦V ₁ V ₄≦1.5P ₁ P ₂ and 0.3g≦S d ₁≦5 2.5g wherein P₁ is a focal point of a paraboloid of the curved surface S₁ and P₂ is a focal point of a paraboloid of the curved surface S₂, and V₁ is a cross point between the Z axis and the curved surface S₁, and V₄ is a cross point between the Z axis and the curved surface S₄, and ^(g) denotes a distance between P₁ and V₁, and d₁ denotes a distance between V₁ and V₂.
 12. A lens according to claim 9, wherein the lens satisfies the following formula 0.3(P ₁ P ₂−2g)≦R _(2≦)3(P ₁ P ₂−2g) wherein P₁ is a focal point of a paraboloid of the curved surface S₁ and P₂ is a focal point of a paraboloid of the curved surface S₂, and V₁ is a cross point between the Z axis and the curved surface S₁, and g denotes a distance between P₁ and V₁, and R₂ is a curvature radius of the curved surface S₂.
 13. A lens according to claim 9, wherein a hyperbolic curve is used instead of a rotational hyperboloid, and wherein a circle is used instead of a spherical surface.
 14. A lens according to claim 9, wherein an approximate spherical surface or an approximate circle is used instead of a rotational hyperboloid or a hyperbolic curve.
 15. A lens according to claim 9, wherein the lens satisfies the following formula $0.5 < {\frac{n_{s}}{n_{0}}} < 1.6$ wherein n_(s) denotes a refractive index of the negative refractive index medium with respect to a vacuum, and n₀ denotes a refractive index of the periphery of the negative refractive index medium with respect to a vacuum.
 16. An optical system, wherein when the lens according to claim 9 is a first lens, in addition to the first lens, there is provided a second lens having two concentric spherical surface formed of a medium indicating negative refraction, and wherein a spherical center of the first lens according to claim 9 substantially coincides with a spherical center of the second lens.
 17. An optical system according to claim 16, wherein thickness of the second lens satisfies the following formula 0.3(P ₁ P ₂−2g)≦R ₂≦3(P ₁ P ₂−2g) wherein P₁ is a focal point of a paraboloid of the curved surface S₁ and P₂ is a focal point of a paraboloid of the curved surface S₂, and V₁ is a cross point between the Z axis and the curved surface S₁, and g denotes a distance between P₁ and V₁, and R₂ is a curvature radius of the curved surface S₂.
 18. An optical system having the lens according to claim 9 and an optical element made of a positive refractive index medium.
 19. An illumination optical system, having the lens according to claim 2, wherein a light source is provided in the vicinity of a focal point of the rotational paraboloid or a parabola.
 20. An illumination optical system according to claim 19, wherein the rotational paraboloid or parabola is approximated by a spherical surface or a circle.
 21. An optical system comprising the lens according to claim 9, wherein an object or a light source is arranged in the vicinity of a focal point of the rotational paraboloid or parabola, rotational hyperboloid or hyperbolic curve, or alternatively, spherical surface or circle.
 22. An optical system comprising the lens according to claim 9, wherein an object or a light source is arranged at a position meeting the following formula $\Delta \leq \frac{f_{1}}{3}$ wherein f₁ is defined as a distance between P₁ and V₁, wherein P₁ is a focal point of a paraboloid of the curved surface S₁ and V₁ is a cross point between the Z axis and the curved surface S₁, and in the vicinity of a focal point of the rotational paraboloid or parabola, rotational hyperboloid or hyperbolic curve, or alternatively, spherical surface or circle.
 23. An image pickup device comprising an image pickup element in addition to the lens according to claim
 4. 24. A lens according to claim 9, wherein the medium indicating negative refraction is a negative refractive index medium.
 25. A lens according to claim 9, wherein the medium indicating negative refraction is a photonic crystal.
 26. A lens according to claim 9, wherein the medium indicating negative refraction is a medium indicating complete image formation property.
 27. A lens according to claim 9, wherein a light beam to be used is a light beam having a single wavelength. 